Is there a way to export an object to Unity 3D while keeping any applied textures?
Textures and diffuse colors are not assigned automatically. You have to manually assign them by dragging the texture onto the mesh in the Scene View in Unity

You don't have to export to .fbx, Unity supports .blend. To import a file into Unity,


Right-click in the Projects pane, choose 'Import new Asset', and navigate to the file
Move the .blend file into your project's Asset folder


You can read more here.